Scale an object that used to be text but keep the same thickness so when I scale up it stays "thin".

To a certain (wobbly) degree you can do that by using symbols with Nine-Slice scaling.

Most of the time some post-editing is required, though.

thejinbins,

Rereading the OP and noticing the SIMPLE EXAMPLE bit, in more complicated cases you will need to use other ways to avoid the shortenings mentioned.

It can be done (at least for some/many cases), the best way depending on the case (the one suggested may be the easiest one for a simple shape).

In the F case shown, the shortening that is inevitable by offsetting can be reversed more or less accurately/tediously by a step like 5) in my suggestion, only moving the outermost Anchor Points away.

thejinbins,

Here is another approach which will keep the (outer) proportions and the thickness, thus avoiding shortening of the slanted ends, unavoidable with Offset Path (this is shown for a slightly different F that is 200 x 400 with a thickness of 20, scaled up 1.6 times to 320 x 640, and still 20):

0) Switch from Fill to Stroke;

1) Scale up the original F;

2) Move a copy to the side and down by the thickness (20 and -20);

3) Select both and Pathfinder>Divide, then Ctrl/Cmd+G to Ungroup;

4) Delete the unwanted bits so you only have the stem/top bar and the floating bar bit;

5) Direct Select the leftmost Anchor Points of the bar bit and move them far enough horizontally (you can use the left Arrow key);

6) Pathfinder>Unite;

7) Remove the unneeded Anchor Point(s, probably only one).

8 ) Switch from Stroke to Fill.

You can use the Shape Builder instead of the Pathfinders, but there is no need.

Click/RightClick to get closer, or Click Here to see in Chrome 

You are scaling a shape, not a stroke, so you can't directly get the result you are looking for. One thing you could try: After scaling up, use a negative Offset Stroke to narrow up the shape. Then delete the outer shape and fill the inner one.